It sounds like one of your extensions, or some custom code in your system, is incompatible with Magento's compilation mode. The exact solution will vary depending on your PHP system. Here's an explanation of the problem and some troubleshooting tips that should help you.
A file in your system contains a line that looks something like this
//the string 'ConfigFileManager-Model.php' may be
//a constant or variable as well
ConfigFileManager-Model.php is not part of Magento, or any publicly distributed package as far as I can tell.
It sounds like when you're running in "non-compiled mode", Magento can find this file without issue. That means it's somewhere in the normal Magento include paths
#possibly more, depending on your system's default include path
However, when Magento runs in compiled mode, the
community code pool paths are not added as include paths. Only the folder
is added as an include path. Additionally, what compilation mode does is make copies of all the class files and drops them into the
includes/src folder. That means the file's own original folder is no longer in the include path. Because of this, when Magento runs the
require code in compilation mode, it can't find the
ConfigFileManager-Model.php file, and fails.
The best thing to do would be recode the module and/or custom code such that the functionality of
ConfigFileManager-Model.php is incorporated into a standard Magento helper of model class.
The less good, but quicker thing would be to identify all the places
required and change the path so it's absolute instead of relative.